arX

iv:h

ep-t

h/98

0616

3v1

19

Jun

1998

CONSISTENT PERTURBATIVE LIGHT-FRONT

FORMULATION OF QUANTUM ELECTRODYNAMICS

Michele Morara

Dipartimento di Fisica ”A. Righi”, Universita di Bologna,

and

Roberto Soldati ♭

Dipartimento di Fisica ”A. Righi”, Universita di Bologna

and Istituto Nazionale di Fisica Nucleare, Sezione di Bologna,

40126 Bologna - Italy

Abstract

A new light-front formulation of Q.E.D. is developed, within the framework of standard

perturbation theory, in which x+ plays the role of the evolution parameter and the gauge

choice is A+ = 0 (light-front ”temporal” gauge). It is shown that this formulation leads to

the Mandelstam-Leibbrandt causal prescription for the non-covariant singularities in the

photon propagator. Furthermore, it is proved that the dimensionally regularized one loop

off-shell amplitudes exactly coincide with the correct ones, as computed within the standard

approach using ordinary space-time coordinates.

PACS numbers : 11.10.Ef, 11.15.Bt, 12.20.-m

DFUB-97-17

April 98

Resubmitted to Phys. Rev. D

♭ E-mail: soldati@bo.infn.it

1

1. Introduction.

Light-front formulation of gauge quantum field theories has become more and more

popular in the last few years. In the abelian case, i.e. standard Q.E.D. or abelian Higgs

model, the renewal of the interest in that subject is mainly because of two reasons. On the

one hand, the light-front hamiltonian approach to Q.E.D. appears to provide an alternative

tool to compute Lamb shift [1] and deal with bound-state problems [2]. On the other hand,

some non-perturbative aspects - such as the role of the zero-modes [3] - have first to be

clearly understood in abelian models, before going into the much more challenging non-

abelian case.

The original attempts to set up canonical quantization of Q.E.D. in the framework of

light-front - or null-plane - dymamics date back to the early seventies [4]. In the original

approach, the light-cone coordinate x+ = (x0 + x3)/√

2 plays the role of the evolution

parameter and the standard gauge choice is A− = 0, in such a way to stay as close

as possible to the axial gauge formulation of Q.E.D. in standard space-time coordinates

(STC).

After a considerable amount of work has been done along this line, it was definitely

discovered [5] that perturbation theory, based upon the original light-front quantization

scheme for gauge theories, is inconsistent, owing to loop integrations, just because the above

scheme necessarily entails the Cauchy principal value (CPV) prescription to understand

the spurious non-covariant poles in the gauge particle vector propagator. This means that

quite basic features of the standard perturbative approach for gauge theories are lost, such

as power counting renormalizability, unitarity, covariance and causality. In other words, the

original approach to light-front quantization of gauge theories is certainly not equivalent

to the standard covariant formulation, already at the perturbative level; it is a fortiori

hard to believe that the same approach could provide useful hints beyond perturbation

theory, in the absence of deep modifications. En passant, it is really curious and rather

surprising that a non-negligeable fraction of the field theorists community seems to have

2

nowadays not yet fully gathered and appreciated this rough breakdown of the conventional

old light-front approach to gauge theories. For instance, even the one loop Q.C.D. beta

function does not result to be, within that context, the correct covariant one [5].

It has been noticed some time ago [6] that, in order to restore at least causality for the

free propagator of the gauge fields in the light-cone gauge, a special prescription, thereof

called the Mandelstam-Leibbrandt (ML) prescription, has to be employed, in order to

regulate the spurious non-covariant singularities. Shortly afterwards, it has been realized

that the ML prescription arises from the canonical quantization in standard STC, provided

some special unphysical (ghost-like) degrees of freedom are taken into account [7]. Even

more, it has been proved that, within that framework, gauge theories in the light-cone

gauge are renormalizable, unitary and covariant order-by-order in perturbation theory [8].

It is worthwhile to emphasize how this remarkable result crucially stems from the presence

of the above mentioned unphysical degrees of freedom: as soon as they are correctly taken

into account, the equivalence between the covariant and light-cone gauges is established,

within the standard perturbative approach in STC.

The open issue, which is still there, is to find a light-front formulation for quantum

gauge theories, which turns out to be equivalent to the conventional one in ordinary STC, at

least in perturbation theory. It is definitely clear, from the above considerations, that such

a new formulation, whatever it is, must lead to the ML prescription for the non-covariant

singularities of the gauge particle vector propagator, at variance with the original old one,

driving instead to the pathological CPV prescription.

A first step towards this direction has been done quite recently by McCartor and

Robertson [9]. They have found an algebraic scheme to quantize the theory on the light-

front, taking also the above mentioned unphysical degrees of freedom into account. How-

ever, as they use the ”temporal” LCC as the evolution parameter and the ”spatial” gauge

choice A− = 0, the above algebraic setting is done after quantization of physical and un-

physical degrees of freedom on different characteristic surfaces, i.e. light-front hyperplanes.

3

Beside being somewhat unnatural * , this approach does not drive exactly to the standard

form of the photon propagator with the ML prescription for the spurious singularity. It is

one of the aims of the present paper to show how the latter drawbacks in the McCartor

and Robertson approach could be indeed overcome, without spoiling its correct content of

an enlarged light-front operator algebra.

In order to achieve this goal, we simply make the transition from the ”spatial” light-

cone gauge A− = 0 to the ”temporal” light-cone gauge A+ = 0, the ”temporal” LCC x+

being kept as the evolution parameter within the light-front formulation. In so doing, on

the one hand the free field operator algebra for the whole set of fields is naturally defined

on the ”spatial” hyperplanes x+ = constant. On the other hand, the ML prescription is

exactly recovered for the propagator of the free radiation field.

These remarkable features allow therefore to correctly develop perturbation theory,

once the corresponding interaction hamiltonian has been single out from constraints anal-

ysis of (pseudo-)classical Q.E.D. in LCC, including unphysical degrees of freedom (i.e. in

an enlarged phase space). This leads to obtain the set of light-front Q.E.D. Feynman’s

rules, which will be shown to involve an infinite set of special non-covariant vertices. It

is then amusing to check, at one loop, that truncated light-front Green’s functions - i.e.

vacuum expectation values of light-front-time ordered product of field operators - are ex-

actly the same as in the usual STC formulation, provided the gauge invariant dimensional

regularization scheme is embodied.

The paper is organized as follows. In Sect.2, we give a critical reading of the McCartor

and Robertson approach to light-front quantization of the free radiation field. In so doing,

we point out where this approach reveals to be unsatisfactory and how to implement it,

in order to reproduce the ML form of the free propagator. In Sect.3 we briefly review the

* Actually, in the presence of interaction, the simultaneous occurrence of ”spatial” and

”temporal” light-front hyperplanes, to specify the operator’s algebra, makes the treatment

somewhat complicated.

4

light-front quantization of the free Dirac’s field, in order to also establish our notations

for the light-front treatment of spinorial matter. In Sect.4 we perform the canonical light-

front quantization of Q.E.D. in the ”temporal” light-cone gauge A+ = 0, by means of the

standard Dirac’s procedure for constrained systems. Sect.5 is devoted to perturbation the-

ory: namely, we derive Feynman’s rules and show that, up to the one loop approximation,

dimensionally regularized truncated and connected light-front Green’s functions are the

same, as computed out of the standard canonical framework in usual STC. Sect.6 contains

some further comments and remarks, as well as an outlook on future developments.

2. Light-front quantization of the free radiation field.

Some time ago [7] it has been shown that the canonical quantization of the free ra-

diation field in the light-cone gauge nµAµ ≡ A− = 0, (n2 = 0), is suitably formulated

using standard space-time coordinates (STC) and leads, eventually, to the ML prescrip-

tion for the spurious singularities in the propagator. It is worthwhile to stress that, in the

derivation of the above result, the unphysical components of the gauge potential play a

fundamental role. On the other hand, within the original approach to light-front quan-

tization using light-cone coordinates (LCC) [4], those unphysical degrees of freedom turn

out to fulfil constraint equations instead of genuine equations of motion. Thereby, they

are eliminated after imposing suitable boundary conditions and, consequently, only the

physical degrees of freedom are indeed submitted to canonical quantization. In so doing,

unfortunately, the spurious singularity in the vector propagator results to be prescribed as

Cauchy principal valued and leads to an inconsistent meaningless perturbation theory.

It is our aim to show in this section how some light-front quantization scheme exists for

the radiation field in LCC, which drives eventually to the ML prescrition for the spurious

singularity of the vector propagator, just like the standard STC formulation does. In order

to achieve this goal, we will develop and improve a recent attempt [9], in which the above

mentioned unphysical components of the gauge potential are retained and quantized in

5

LCC according to a new procedure. Let us first briefly review the main points of this

approach.

The starting point is the lagrangean density of the free radiation field

Lrad = −1

4FµνF

µν − ΛnµAµ , (2.1)

where nµ = (n+, n⊥, n−) = (1, 0, 0, 0), in such a way that nµAµ = A−, and Λ is a Lagrange

multiplier which enforces the gauge constraint.

The Euler-Lagrange equations lead to

∂µFµν = nνΛ , (2.2a)

A− = 0 . (2.2b)

It is convenient to introduce some new field variables as follows: namely,

Aα = Tα − ∂α∂2⊥

ϕ , (2.3a)

A+ =∂α∂−

Tα − ∂+

∂2⊥

ϕ − 1

∂2⊥

Λ ; (2.3b)

then Eq.s (2.2) become

(2∂+∂− − ∂2⊥)Tα = 0 , (2.4a)

∂−ϕ = ∂−Λ = 0 . (2.4b)

We notice that, as the fields Tα(x) fulfil free D’Alembert’s equations of motion, then

the inverse of the light-front-space derivative in eq. (2.3b) is understood here to be (1/∂−) ≡

(2∂+/∂2⊥). Furthermore, from eq. (2.4b) we can easily see that the fields ϕ and Λ do not

fulfil evolution equations - remember that here it is the LCC x+ which plays the role of

the evolution parameter - but, as previously noticed, they satisfy constraint equations and,

therefrom, can not be canonically quantized on the null hyperplanes at constant x+.

Now, it has been suggested [9], [10] a new light-front quantization procedure, in which

the transverse fields Tα are quantized on null hyperplanes at equal x+, according to the

6

original light-front recipe, while the longitudinal fields ϕ and Λ at equal x−. Following

this procedure, one can set up the generators of the translations on the null hyperplanes

Σ+ and Σ−, in the limit L→ ∞ (see Fig.1), and obtain, taking the Heisenberg equations

of motion (2.4) into account, the commutation relations

[Tα(x), Tβ(y)]x+=y+ = − i

2δαβδ

(2)(x⊥ − y⊥)sgn(x− − y−) , (2.5a)

[ϕ(x),Λ(y)]x−=y− = iδ(x+ − y+)∂2⊥δ

(2)(x⊥ − y⊥) , (2.5b)

[Tα(x), ϕ(y)] = [Tα(x),Λ(y)] = [ϕ(x), ϕ(y)] = [Λ(x),Λ(y)] = 0 , (2.5c)

where sgn(x) denotes the usual sign distribution. In so doing, the Authors of Ref.s [9]

suggest that the light-cone-time ordered product of the gauge potential operators defined

by

D+µν(x− y) ≡ θ(x+ − y+) 〈0|Aµ(x)Aν(y)|0〉+ θ(y+ − x+) 〈0|Aν(y)Aµ(x)|0〉 , (2.6)

might eventually give rise to the ML form of the gauge field propagator. Actually we shall

show below that this is not exactly true, owing to the presence of some ill-defined products

of tempered distributions.

7

As a matter of fact, if we consider the longitudinal components of the gauge potential:

namely,

Γµ = − 1

∂2⊥

(∂µϕ+ nµΛ) , (2.7)

then a straightforward calculation yields

〈0|Γµ(x)Γν(y)|0〉 =

∫

d4k

(2π)3eik(x−y)θ(−k+)δ(k−)

nµkν + nνkµk2⊥

. (2.8)

After multiplication, for instance, with θ(x+ − y+) and taking the Fourier transform we

formally get the convolution

∫ 0

−∞

dξ

2πi

δ(k−)

(k+ − ξ) − iǫ

[

nµkν + nνkµk2⊥

]

k+=ξ

.

One can easily convince himself that the above expression does not define a tempered

distribution - owing to the logarithmic divergence in the ξ-integration - which means, in

turn, that the propagator in eq. (2.6) is not properly understood from the mathematical

point of view.

Nonetheless, it is indeed remarkable that the main idea behind the quantization pro-

cedure in Ref.s [9], i.e. the enlarged algebra on the characteristic surfaces in order to fulfil

causality, is suggestive, albeit troubles arise when dealing with the evolution. It should be

apparent that, in fact, the very same reasons preventing us from specifying the algebra of

the longitudinal field operators at equal x+, also prevent us from propagating the unphys-

ical degrees of freedom along x+. The simplest way to circumvent these difficulties and

to build up a consistent light-front dynamics turns out to be a change of the null gauge

vector♯, i.e. we replace nµ 7−→ n∗µ ≡ (0, 0, 0, 1) in such a way that n∗µAµ = A+ = 0.

♯ Actually, an equivalent way to proceed is to keep the previous light-cone gauge choice

unaltered and to change the evolution parameter (the light-front-time) from x+ to x−,

the key point being that the light-front-time and the light-cone gauge vector have to be

parallel.

8

Let us therefore consider the new lagrangean density

Lrad = −1

4FµνF

µν − Λn∗µAµ ; (2.9)

as the whole set of fields now fulfils genuine equations of motion, it is convenient to proceed

within the framework of Dirac’s canonical quantization [11].

The canonical momenta are ( Arad ≡∫

d4xLrad(x) )

π− ≡ δArad

δ∂+A−

= F+− , (2.10a)

πα ≡ δArad

δ∂+Aα= F−α , (2.10b)

π+ ≡ δArad

δ∂+A+= 0 , (2.10c)

πΛ ≡ δArad

δ∂+Λ= 0 , (2.10d)

whence it follows that there are two primary second class constraints (2.10b) originating

from the use of LCC, as well as two primary first class constraints (2.10c-d).

The canonical hamiltonian becomes

Hrad =

∫

d3x

{

1

2

(

π−)2

+1

4FαβFαβ − A+

(

∂απα + ∂−π

− − Λ)

}

, (2.11)

and, consequently, from the light-front-temporal consistency of the first class constraints

(2.10c-d) we derive the secondary constraints

A+ = 0 , (2.12a)

∂απα + ∂−π

− − Λ = 0 . (2.12b)

The full set of constraits is now second class and thereby we can compute the Dirac’s

brackets. After choosing as independent fields the following ones,

φ1 = A1 , φ2 = A2 , φ3 = A− , φ4 = π− , (2.13)

9

we eventually obtain the Dirac’s brackets matrix

Φab(x,y) ≡ {φa(x), φb(y)}D|x+=y+ , a, b = 1, 2, 3, 4 ,

whose matrix elements are integro-differential operators in terms of light-front-space coor-

dinates x = (x1, x2, x−): namely,

Φab(x,y) =

∣

∣

∣

∣

∣

∣

∣

−1/2∂− 0 0 ∂1/2∂−0 −1/2∂− 0 ∂2/2∂−0 0 0 1

−∂1/2∂− −∂2/2∂− −1 ∂2⊥/2∂−

∣

∣

∣

∣

∣

∣

∣

. (2.14)

Here the identity 1 means the product δ(x−−y−)δ(2)(x⊥−y⊥), whilst the kernels (1/∂−)

and (∂α/∂−) are shorthands for δ(2)(x⊥−y⊥)sgn(x−−y−) and(

∂αδ(2)

)

(x⊥−y⊥)sgn(x−−

y−) respectively. It should be noticed that the sign distribution is such to enforce standard

(anti-)symmetry properties of Dirac’s brackets.

After setting the secondary constraints strongly equal to zero in the hamiltonian

(2.11), we obtain the Dirac’s form

HD =

∫

d3x

{

1

2

(

π−)2

+1

4FαβFαβ

}

. (2.15)

Now, in order to simplify the equations of motion, it is convenient to make the change

of variables similar to the one of Eq.s (2.3) but taylored to the present light-cone gauge

choice A+ = 0: namely,

Aα = Tα − ∂α∂2⊥

ϕ , (2.16a)

A− =2∂−∂2⊥

∂αTα − ∂−∂2⊥

ϕ − 1

∂2⊥

Λ ; (2.16b)

π− = ∂αTα . (2.16c)

The Dirac’s brackets among the new independent fields read

Φ′ab(x,y) =

∣

∣

∣

∣

∣

∣

∣

−1/2∂− 0 0 00 −1/2∂− 0 00 0 0 ∂2

⊥

0 0 −∂2⊥ 0

∣

∣

∣

∣

∣

∣

∣

. (2.17)

10

where we have set

φ′1 = T1 , φ′2 = T2 , φ

′3 = ϕ , φ′4 = Λ , (2.18)

and the canonical hamiltonian takes its final Dirac’s form

H ′D ≡

∫

d3x

{

1

2∂βTα∂βTα

}

, (2.19)

whence we obtain the genuine equations of motion

∂+Tα =∂2⊥

2∂−Tα , (2.20a)

∂+ϕ = 0 , (2.20b)

∂+Λ = 0 . (2.20c)

The transition to the quantum theory is accomplished under replacement of the Dirac’s

brackets with canonical equal light-front-time commutation relations, which read

[Tα(x), Tβ(y)]x+=y+ = − i

2δαβδ

(2)(x⊥ − y⊥)sgn(x− − y−) , (2.21a)

[ϕ(x),Λ(y)]x+=y+ = iδ(x− − y−)∂2⊥δ

(2)(x⊥ − y⊥) , (2.21b)

[Tα(x), ϕ(y)] = [Tα(x),Λ(y)] = [ϕ(x), ϕ(y)] = [Λ(x),Λ(y)] = 0 . (2.21c)

It is important to notice that the above canonical commutation relations (CCR) have the

very same form as in the McCartor and Robertson quantization scheme, see Eq.s (2.5), up

to the crucial difference that now the quantization characteristic surface is the same for all

the fields.

Let us now search for the solutions, in the framework of the tempered distributions,

of the equations of motion in the Fourier space. To this aim, it is convenient to introduce

again the longitudinal (unphysical) components of the radiation field

Γµ = − 1

∂2⊥

(

∂µϕ+ n∗µΛ

)

, (2.22)

in such a way that

Tµ(x) ≡ Aµ(x) − Γµ(x) . (2.23)

11

For the transverse components we easily get

Tµ(x) =

∫

d2k⊥dk−(2π)3/2

θ(k−)√

2k−εαµ(k⊥, k−)

{

aα(k⊥, k−)e−ikx + a†α(k⊥, k−)eikx}

k+=k2⊥/2k−

,

(2.24)

where the (real) polarization vectors are given by

ε1µ(k⊥, k−) =

∣

∣

∣

∣

∣

∣

∣

010

2k1k−/k2⊥

∣

∣

∣

∣

∣

∣

∣

, ε2µ(k⊥, k−) =

∣

∣

∣

∣

∣

∣

∣

001

2k2k−/k2⊥

∣

∣

∣

∣

∣

∣

∣

, (2.25)

whilst the longitudinal components read

Γµ(x) =

∫

d2k⊥dk−(2π)3/2

θ(k−)√k⊥

{[

− kµk⊥

f(k⊥, k−) + n∗µg(k⊥, k−)

]

e−ikx + h. c.

}

k+=0

,

(2.26)

where k⊥ ≡√

k21 + k2

2 . The canonical commutation relations (2.21) entail the following

algebra of the creation-annihilation operators: namely,

[

aα(k⊥, k−), a†β(p⊥, p−)]

= δαβδ(2)(k⊥ − p⊥)δ(k− − p−) , (2.27a)

[

f(k⊥, k−), g†(p⊥, p−)]

= δ(2)(k⊥ − p⊥)δ(k− − p−) , (2.27b)

[

g(k⊥, k−), f †(p⊥, p−)]

= δ(2)(k⊥ − p⊥)δ(k− − p−) , (2.27c)

all the other commutators vanishing.

The canonical commutation relations (2.27b-c) show that the theory involves an in-

definite metric space of states. The physical subspace Vphys, whose metric turns out to be

positive semi-definite, is defined through the condition [7]:

g(k⊥, k−) |v〉 = 0 , ∀ |v〉 ∈ Vphys . (2.28)

It should be noted that, as

〈w|Λ(x)|v〉 = 0 , ∀ |w〉 , |v〉 ∈ Vphys , (2.29)

the Gauss’ law is indeed fulfilled in Vphys.

12

Let us finally compute the free vector propagator

D+µν(x− y) ≡ θ(x+ − y+) 〈0|Aµ(x)Aν(y)|0〉+ θ(y+ − x+) 〈0|Aν(y)Aµ(x)|0〉 , (2.30)

which, after the gauge fixing condition (2.12a), turns out to be properly defined from

the mathematical point of view, i.e. the product of the distributions in eq. (2.30) does

indeed exist. Separating the transverse and longitudinal components, setting aµν(k) ≡

n∗µkν + n∗

νkµ and going to the momentum space we eventually get

DTµν(k) =

i

k2 + iǫ

[

−gµαgαν +2k−k2⊥

aµν(0, k⊥, k−)

]

, (2.31)

DΓµν(k) = −i k−

k+k− + iǫ

aµν(0, k⊥, k−)

k2⊥

. (2.32)

Taking into account that

2k−k2⊥

(

1

k2 + iǫ− 1

2k−k+ + iǫ

)

=1

k2 + iǫ

1

[k+], (2.33)

where

1

[k+]≡ 1

k+ + iǫsgn(k−)≡ k−k−k+ + iǫ

, (2.34)

which is nothing but the Mandelstam-Leibbrandt distribution, we finally get the propaga-

tor in the momentum space

D+µν(k) =

i

k2 + iǫ

[

−gµν +n∗µkν + n∗

νkµ

[n∗k]

]

. (2.35)

It has to be stressed that, more than being mathematically well defined, the present form

of the free vector propagator exactly coincides with the one obtained in the framework of

ordinary time canonical quantization of Ref. [7]. This means that the light-front operator

algebra (2.21) together with light-front-time propagation are completely equivalent, at

the level of the free field theory, to the ordinary time canonical quantization and standard

chronological pairing, at variance with the old light-front formulation of Ref.s [4]. This non

trivial result, which arises as the correct implementation of the original ideas of Ref.s [9],

13

will survive after the switching on of the interaction with spinor matter, as we shall discuss

below.

3. Light-front quantization of the free Dirac field.

Before going to the treatment of Q.E.D. it is useful to briefly review the canonical

light-front quantization of the free Dirac field and, in so doing, establish our conventions

and notations. First we recall that, in order to obtain the correct canonical anticom-

mutation relations from Dirac’s procedure, it is convenient to consider the system at the

(pseudo)classical level. This means that we start from spinor fields in terms of Grassmann-

valued fields satysfying the graded version of the canonical Poisson’s and Dirac’s brackets

(see, for instance Ref. [12]). The same formalism will be generalized in the next Section,

where Bose fields are also included.

Within the framework of the light-front quantization, it is customary to introduce the

following representation of the Dirac’s matrices: namely,

γ+ =

∣

∣

∣

∣

0 0√2σ1 0

∣

∣

∣

∣

γ1 =

∣

∣

∣

∣

−iσ2 00 −iσ2

∣

∣

∣

∣

γ2 =

∣

∣

∣

∣

iσ1 00 −iσ1

∣

∣

∣

∣

γ− =

∣

∣

∣

∣

0√

2σ1

0 0

∣

∣

∣

∣

, (3.1)

and we write the four-component Dirac’s spinor as

Ψ ≡∣

∣

∣

∣

ψχ

∣

∣

∣

∣

, (3.2)

with ψ, χ two-components complex spinors. Here σi, i = 1, 2, 3 are the Pauli’s matrices

and we also set

τ1 ≡ σ3 , τ2 ≡ i12 . (3.3)

Therefore, the lagrangean density for the free Dirac’s field

LD = Ψ (iγµ∂µ −m) Ψ , (3.4)

where Ψ ≡ Ψ†γ0, γ0 = 2−1/2(γ+ + γ−), may be rewritten as

LD = ψ†i√

2∂+ψ + χ†i√

2∂−χ+ ψ†(

iτα†∂α −mσ1)

χ+ χ†(

iτα∂α −mσ1)

ψ , (3.5)

14

whence the canonical momenta read

πψ = −i√

2ψ† , (3.6a)

πψ†

= 0 , (3.6b)

πχ = 0 , (3.6c)

πχ†

= 0 . (3.6d)

It follows that we have two primary second class constraints (3.6a-b) and two primary

first class constraints (3.6c-d). The canonical hamiltonian turns out to be

H =

∫

d3x{

−χ†i√

2∂−χ− ψ†(

iτα†∂α −mσ1)

χ− χ†(

iτα∂α −mσ1)

ψ}

(3.7)

and the light-front-temporal consistency of the first class constraints lead to the onset of

the secondary constraints

i√

2∂−χ† + i∂αψ

†τα† +mψ†σ1 = 0 , (3.8a)

i√

2∂−χ+(

iτα∂α −mσ1)

ψ = 0 . (3.8b)

Now, the whole set of constraints being second class, the graded Dirac’s bracket can

be consistently defined and taking ψ and ψ† as independent fields we readily find

{

ψr(x), ψ†r′(y)

}

D

∣

∣

∣

x+=y+

=1√2δrr′δ

(2)(x⊥ − y⊥)δ(x− − y−) , r, r′ = 1, 2 , (3.9)

all the other graded Dirac’s brackets vanishing.

After solving the secondary constraints (3.8) in terms of the independent fields ψ, ψ†

the canonical hamiltonian (3.7) can be cast into Dirac’s form: namely,

HD = i√

2

∫

d3x

{

ψ† ∂2⊥ −m2

2∂−ψ

}

, (3.10)

from which we obtain the canonical equations of motion

∂+ψr =∂2⊥ −m2

2∂−ψr , (3.11a)

∂+ψ†r =

∂2⊥ −m2

2∂−ψ†r , (3.11a)

15

showing that the independent fields ψ, ψ† correctly fulfil the Klein-Gordon equation.

The expansion into normal modes leads to the standard decomposition

ψ(x) =

∫

d3k

(2π)3/2θ(k−)

21/4

×∑

s=±1/2

{

wsbs(k⊥, k−)e−ikx + w−sd†s(k⊥, k−)eikx}

k+=(k2⊥

+m2)/2k−,

(3.12)

and hermitean conjugate, where the polarization vectors are simply given by

ws=1/2 ≡∣

∣

∣

∣

10

∣

∣

∣

∣

, ws=−1/2 ≡∣

∣

∣

∣

01

∣

∣

∣

∣

. (3.13)

As it is well known the graded Dirac’s brackets (3.9) entail the canonical operator

algebra

{

bs(k⊥, k−), b†s′(p⊥, p−)}

= δss′δ(2)(k⊥ − p⊥)δ(k− − p−) , (3.14a)

{

ds(k⊥, k−), d†s′(p⊥, p−)}

= δss′δ(2)(k⊥ − p⊥)δ(k− − p−) , (3.14b)

all the other anticommutators vanishing.

We are now ready to compute the free light-front fermion propagator which is defined

to be

iS+(x− y) ≡ θ(x+ − y+)⟨

0|Ψ(x)Ψ(y)|0⟩

− θ(y+ − x+)⟨

0|Ψ(y)Ψ(x)|0⟩

. (3.15)

To this aim, it is convenient to introduce the light-front pairing between two-components

spinors αr, βr′ , r, r′ = 1, 2, in such a way that

S+αβrr′ (x− y) ≡ θ(x+ − y+) 〈0|αr(x)βr′(y)|0〉 − θ(y+ − x+) 〈0|βr′(y)αr(x)|0〉 ; (3.16)

then the propagator (3.15) can be cast into a matrix form: namely,

iS+(x− y) =

∣

∣

∣

∣

S+ψχ†

σ1 S+ψψ†

σ1

S+χχ†

σ1 S+χψ†

σ1

∣

∣

∣

∣

. (3.17)

The only independent light-front pairing turns out to be

S+ψψ†

(x− y) = τ2√

2∂−D(x− y;m) , (3.18)

16

where

D(x− y;m) =

∫

d4k

(2π)4i

k2 −m2 + iǫeik(x−y) (3.19)

is the free propagator of the massive real scalar field. Then, from the constraints (3.8) we

eventually obtain

S+ψχ†

(x− y) =(

iτα†∂α +mσ1)

D(x− y;m) , (3.20a)

S+χψ†

(x− y) =(

iτα∂α +mσ1)

D(x− y;m) , (3.20b)

S+χχ†

(x− y) = τ2√

2∂2⊥ −m2

2∂−D(x− y;m)

= τ2√

2∂+D(x− y;m) + iτ2 1√2∂−

δ(4)(x− y) . (3.20c)

As a consequence, from Eq. (3.17) and taking Eq. (3.1) into account, the free fermion

light-front propagator can be written in the form

iS+(x− y) = (iγµ∂µ +m)D(x− y;m) − γ+

2∂−δ(4)(x− y) , (3.21)

where the first term in the RHS is the usual covariant fermion propagator

Scov(x− y) =

∫

d4k

(2π)4m− γµkµk2 −m2 + iǫ

eik(x−y) , (3.22)

whilst the second one is the so called ”instantaneous” or ”contact” term, which is generated

by the propagation along the light-cone generating lines. The role of those term will be

further elucidated in the next sections; in particular, it will be clear that there is no need to

specify any prescription to define the light-front-space anti-derivative ∂−1− which appears

in eq. (3.21).

4. Light-front Q.E.D. in the light-cone temporal gauge.

We are now ready to discuss the main subject, i.e. the perturbative light-front for-

mulation of spinor Q.E.D., in which the LCC x+ plays the role of evolution parameter,

within the light-cone gauge choice A+ = 0. Owing to this pattern (the controvariant LCC

17

x+ just corresponds to the covariant component A+ of the abelian vector potential), this

formulation will be naturally referred to as light-front Q.E.D. in the light-cone temporal

gauge.

The starting point is obviously the lagrangean density

L = −1

4FµνF

µν − ΛA+ + Ψ (iγµ∂µ −m)Ψ + eAµΨγµΨ , (4.1)

which can be rewritten, using the notations of the previous section, in the form

L = −1

4FµνF

µν − ΛA+

+ ψ†i√

2∂+ψ + χ†i√

2∂−χ+ ψ†(

iτα†∂α −mσ1)

χ+ χ†(

iτα∂α −mσ1)

ψ

+ eA+

√2ψ†ψ + eAα

(

ψ†τα†χ+ χ†ταψ)

+ eA−

√2χ†χ .

(4.2)

As the interaction does not contain derivative couplings, the definitions of the canon-

ical momenta do not change with respect to the free case: then we have,

π− = F+− , (4.3a)

πα = F−α , (4.3b)

π+ = 0 , (4.3c)

πΛ = 0 , (4.3d)

πψ = −i√

2ψ† , (4.3e)

πψ†

= 0 , (4.3f)

πχ = 0 , (4.3g)

πχ†

= 0 . (4.3h)

where, again, (4.3b-e-f) are primary second class constraints whilst the remaining ones,

but eq. (4.3a), are primary first class. The canonical hamiltonian reads

H =

∫

d3x

{

1

2

(

π−)2

+1

4FαβFαβ −A+

(

∂απα + ∂−π

− − Λ)

− χ†i√

2∂−χ− ψ†(

iτα†∂α −mσ1)

χ− χ†(

iτα∂α −mσ1)

ψ

−eA+

√2ψ†ψ − eAα

(

ψ†τα†χ+ χ†ταψ)

+ eA−

√2χ†χ

}

(4.4)

18

and from the light-front temporal consistency of the primary first class constraints the

following secondary constraints arise: namely,

A+ = 0 , (4.5a)

∂απα + ∂−π

− − Λ − e√

2ψ†ψ = 0 , (4.5b)

i√

2D∗−χ

† + iD∗αψ

†τα† +mψ†σ1 = 0 , (4.5c)

i√

2D−χ+(

iταDα −mσ1)

ψ = 0 , (4.5d)

where, as usual, we have set Dµ ≡ ∂µ − ieAµ.

The whole set of primary and secondary constraints is now second class and we can

proceed to the calculation of graded Dirac’s brackets. To this aim, however, it is better

to make a preliminary observation. From the constraint equations (4.5c-d) it is apparent

that, if we want to express the two-components spinors χ and χ† as functionals of the

independent ones ψ and ψ†, we have to invert the differential operator D− = ∂− − ieA−.

In the present context the corresponding Green’s function will be understood as a formal

series: namely,

1

D−

≡ 1

∂−

∞∑

n=0

(

ieA−

1

∂−

)n

, (4.6)

where each anti-derivative acts upon all the factors on its right.

As it will be clear later on, we remark that it is neither necessary nor convenient

to specify any kind of prescription, in order to properly define the anti-derivative itself.

Furthermore, it is unavoidable that the Dirac’s hamiltonian, in which all the constraints

are solved in terms of the independent fields, would result into a formal (infinite) power

series of the dimensionless electric charge e.

Let us turn now to the calculation of the graded Dirac’s brackets. As the actual

inversion of the constraints matrix is a little bit complicated in the present case, it is

19

convenient to operate iteratively and compute some sequences of preliminary brackets

(eventually four sequences). After taking

ξ1 ≡ A1, ξ2 ≡ A2, ξ3 ≡ A−, ξ4 ≡ π−, ξ5 ≡ ψ, ξ6 ≡ ψ†, (4.7)

as independent fields, a straightforward although very tedious calculation leads to the

following result: namely,

Ξab(x,y) =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

−1/2∂− 0 0 ∂1/2∂− 0 00 −1/2∂− 0 ∂2/2∂− 0 00 0 0 1 0 0

−∂1/2∂− −∂2/2∂− −1 ∂2⊥/2∂− 0 0

0 0 0 0 0 −i/√

20 0 0 0 −i/

√2 0

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

, (4.8)

where, once again, we have denoted the Dirac’s brackets matrix as

Ξab(x,y) ≡ {ξa(x), ξb(y)}D|x+=y+ , a, b = 1, . . . , 6 .

It is important to realize that the set of the independent interacting fields ξa(x), a =

1, . . . , 6, do obey the very same algebra as the corresponding independent free fields,

notwithstanding the fact that the secondary constraints are quite different in the two

cases. This feature, as we shall see in the sequel, is of crucial importance in setting up the

perturbation theory. Moreover, it has to be gathered that the above property does not

hold in general for an arbitrary constrained system, but it depends, in the present case,

upon a clever choice of the independent fields.

Finally, after solving the secondary constraints in terms of the independent fields

ξa(x), a = 1, . . . , 6, the hamiltonian (4.4) takes its Dirac’s form which becomes

HD =

∫

d3x

{

1

2

(

π−)2

+1

4FαβFαβ

−(

i∂αψ†τα† +mψ†σ1 − eAαψ

†τα†) 1

i√

2D−

(

iτα∂αψ −mσ1ψ + eAαταψ

)

}

,

(4.9)

which is the starting point to develop perturbation theory as we discuss in the next section.

20

5. Perturbation theory.

In order to separate the interaction hamiltonian in a constrained system, one has to

be very careful in the choice of the independent canonical variables: as a matter of fact,

the basic criterion to select the latter ones is eventually dictated by the structure of the

Dirac’s brackets of the interacting theory.

On the one hand, after choosing ξa(x), a = 1, . . . , 6 as independent fields, we see

that the first line of the RHS of Eq. (4.9) does not contain the coupling constant e and,

consequently, does not contribute to the interaction hamiltonian. On the other hand,

had we chosen as independent fields the set A1, A2, A−, Λ, ψ, ψ†, which is a perfectly

legitimate choice, then, after solving π− as a functional of the above variables, we find that

the first line in the RHS of Eq. (4.9) does indeed contribute to the interaction hamiltonian

through the two terms:

−e√

2

{

−∂αAα +1

∂−

(

∂2⊥A− + Λ

)

}

1

∂−

(

ψ†ψ)

+ e2{

1

∂−

(

ψ†ψ)

}2

,

whence, thereby, a quite different kind of perturbation theory does follow.

In view of the above remark, one could be eventually led to the conclusion that

perturbation theory for constrained systems is not univocally determined, owing to the

fact that it depends upon the specific choice of the independent fields, in terms of which

the constraints are solved. Actually, this apparent ambiguity is not there. As a matter

of fact, we recall that perturbation theory stems from the assumption of the existence, at

least formally, of the so called evolution operator, which implements the time-dependent

unitary transformation relating the interacting to the free fields - see, for instance, [12].

On the other hand, we know that a unitary operator is such to preserve the canonical

equal time field algebra. This means that, in the case of constrained systems, the suitable

independent interacting fields must fulfil the very same equal time operator algebra as the

corresponding free fields do. In terms of those, and only those, independent interacting

21

fields the interaction hamiltonian has to be expressed and perturbation theory will be

safely and consistently developed .

From the constraints (4.3b), (4.5b) and the Dirac’s brackets (4.8), it is an easy exercise

to show that

{Λ(x), ψ(y)}D|x+=y+ = ieψ(x)δ(3)(x− y) , (5.1)

whereas, in the free field case, the corresponding Dirac’s bracket vanishes. As a con-

sequence, the construction of the interacting hamiltonian as a functional of the fields

A1, A2, A−, Λ, ψ, ψ† does not make sense in order to set up perturbation the-

ory. The interaction hamiltonian is expressed in terms of the set of independent fields

ξa(x), a = 1, . . . , 6, whose Dirac’s brackets (4.8) do not depend upon the electric charge

e, what makes it now clear why the above algebra (4.8) has been precisely put forward.

We now consider the second line of the hamiltonian (4.9). As all the field operators

in the interaction picture evolve according to free equations of motion, it is convenient to

replace with χ and χ† those linear combinations of the fields ψ and ψ†, which coincide

with the solutions of the free constraint equations (3.8a-b). After this, we can rewrite the

hamiltonian (4.9) in the form:

HD =

∫

d3x

{

1

2

(

π−)2

+1

4FαβFαβ

+(

i√

2∂−χ† + eAαψ

†τα†) 1

i√

2D−

(

−i√

2∂−χ+ eAαταψ

)

}

.

(5.2)

If we now perform, within the second line of the above equation, the following replace-

ments: namely,

χ 7−→ 1

2γ+γ−Ψ , (5.3a)

ταψ 7−→ 1√2γ+γαΨ , (5.3b)

χ† 7−→ 1√2Ψγ− , (5.3c)

ψ†τα† 7−→ 1

2Ψγαγ+γ− , (5.3d)

22

we eventually obtain

HD =

∫

d3x

{

1

2

(

π−)2

+1

4FαβFαβ + Ψγ−i∂−Ψ − eAµΨγ

µΨ +e2

2AµΨγ

µ 1

iD−

AνγνΨ

}

.

(5.4)

It is evident, from the above final form of the Dirac’s hamiltonian, that the interaction

hamiltonian density, upon which perturbation theory is set, reads

Hint = −eAµΨγµΨ +e2

2AµΨγ

µ γ+

iD−

AνγνΨ . (5.5)

It is now apparent that, besides the usual covariant vertex of Q.E.D., we have to consider,

taking the formal definition (4.6) into account, an infinite number of non-covariant vertices.

On the other hand, we have seen that also the free Dirac’s propagator (3.21) exhibits a

non-covariant term besides the usual one. What happens, as we shall here explicitely show

up to the one loop order, is that in dimensionally regularized truncated Green’s functions

all those non-covariant terms cancel, leaving us with the very same renormalizable one

loop structures, as found in the standard STC framework [8].

To this aim, let us first obtain the Feynman’s rules. From the definition (4.6) together

with the identity

γ+

2A− =

γ+

2Aνγ

ν γ+

2, (5.6)

we can formally expand the interaction hamiltonian density as

iHint = ieAµΨγµΨ

− ieAµΨγµ γ

+

2∂−ieAνγ

νΨ

− ieAµΨγµ γ

+

2∂−ieAργ

ρ γ+

2∂−ieAνγ

νΨ

− ieAµΨγµ γ

+

2∂−ieAργ

ρ γ+

2∂−ieAσγ

σ γ+

2∂−ieAνγ

νΨ + . . . ,

(5.7)

where the anti-derivatives (integral operators) act upon all the factors on their right.

From eq.s (2.35), (3.21) and (5.7), we get the Feynman’s rules listed in Fig.2.

23

24

Using these rules, it is not difficult to check graphically that in the one loop truncated

Green’s functions, but photon self-energy diagram, all the non-covariant terms cancel al-

gebraically.

For instance, taking two covariant vertices (Fig.2e) and a second order non-covariant

one (Fig.2f), we reconstruct the full one loop electron self-energy (see Fig.3a), which, after

the removal of the external legs, turns out to be the correct renormalizable one of the

standard STC approach.

Moreover, the one loop renormalizable electron-positron-photon proper vertex can be

reconstructed (see Fig.3b) taking the covariant vertices of Fig.2e as well as first and second

order non-covariant vertices of Fig.s 2f-g into account.

Let us come now to the photon one loop self-energy of Fig.3c.

25

After summation of the relevant vertices, we see that, beside the correct standard

diagram, a further non-covariant graph is there, whose corresponding integral (in 2ω space-

time dimensions) is provided by:

Iρσ(p−) = (ie)2∫

d2ωl

(2π)2ωTr

{

γργ+

2i(l− + p−)γσ

γ+

2il−

}

. (5.8)

However, since∫

d2ωl =

∫

dl+

∫

dl−

∫

d2ω−2l⊥ ,

we immediately see that integration over transverse momenta in (5.8) gives a vanishing

result. This is the only point, up to the one loop approximation, in which the cancellation

of non-covariant vertices does not take place algebraically but involves a further analytic

tool. Owing to the above cancellation mechanisms, either algebraic or due to dimensional

regularization of integrals over transverse momenta, it becomes clear why it is immaterial to

specify any prescrition to understand non-covariant denominators in fermions propagators

as well as in the interaction vertices, at least in perturbation theory.

To sum up, we have shown that, concerning one loop dimensionally regularized trun-

cated Green’s functions, the light-front formulation of Q.E.D. in the light-cone temporal

26

gauge actually reproduces the very same result as in the standard STC renormalizable and

(perturbatively) unitary approach [8], in which non-covariant singularities are regulated

by means of the ML prescription.

6. Conclusion.

A consistent light-front formulation of perturbative Q.E.D. has been worked out in the

light-cone gauge A+ = 0, in which the LCC x+ plays the role of the evolution parameter.

Owing to this, it is natural, by analogy with the ordinary STC formulation, to refer our

choice as to ”temporal” light-cone gauge, alternative to the original ”axial” choice A− = 0.

By consistent, we understand that the quantization scheme here developed reproduces,

at least up to the one loop order, the same off-shell amplitudes as computed from the

conventional correct approach in usual STC [7], [8], which embodies the ML prescription

to define the spurious non-covariant singularities of the free photon propagator.

This result is non-trivial and, in turn, also rather surprising. As a matter of fact, it

has been thoroughly unravelled [14] that in the quantization of gauge theories in ordinary

STC, the use of the temporal (or Weyl) subsidiary condition A0 = 0 is undoubtely much

more troublesome than the axial one A3 = 0, which is in turn also affected by subtle

mathematical pathologies [15]. Eventually, in spite of the huge number of attempts and

efforts, the problem of setting up a fully consistent perturbation theory in the temporal

gauge is still to be solved.

On the contrary, within the light-front perturbative formulation, the ”temporal” gauge

choice A+ = 0 appears to be the safe one, which naturally leads to the ML prescrition

and thereby to the equivalence with the convention approach in STC, whilst the ”spatial”

choice A− = 0 drives to inconsistency [5].

A further comment is deserved to the gauge invariace of the regularization methods in

perturbation theory. It clearly appears that, in the present context, the use of dimensional

regularization is crucial, in order to provide an infinite set of diagrams cancellation, in the

27

absence of which gauge invariance of Q.E.D. would be lost. Things are not so lucky for

cut-off or Pauli-Villars regularizations, which, thereof, turn out to be quite inconvenient

within the perturbative light-front approach.

It should be noticed that the presence of an infinite number of non-covariant vertices,

switching on order-by-order in light-front perturbation theory, closely figures the structure

of counterterms for the 1PI-vertices in the standard STC approach to the light-cone gauges

[8]. This feature is probably connected to the specific properties of the ML propagator,

i.e. to the kind of structures it generates after loop integrations.

Although graphically transparent, a formal general proof - which is basically by in-

duction - that the cancellation mechanism for non-covariant terms persists, to all order in

perturbation theory, will be presented in a forthcoming paper, together with the general-

ization of the present treatment to the non-abelian case.

Acknowledgments.

One of us (R. S.) warmly thanks G. McCartor and D. G. Robertson for quite useful

conversations, correspondence and discussions and is grateful to G. Nardelli for reading

the manuscript. This work has been supported by a MURST grant, quota 40%.

References

[1] B. D. Jones, R. J. Perry : Phys. Rev. D55 (1997) 7715; M. M. Brisudova, R. J.

Perry : Phys. Rev. D54 (1996) 6453.

[2] B. D. Jones : hep-th/9703106 ; B. D. Jones, R. J. Perry, S. D. Glazek : Phys. Rev.

D55 (1997) 6561; M. Krautgartner, H. C. Pauli, F. Wolz : Phys. Rev. D45 (1992)

3755; J. R. Hiller : ”Theory of Hadrons and Light-Front QCD”, World Scientific,

Singapore (1995) 277-281.

[3] A. C. Kalloniatis, D. G. Robertson : Phys. Rev. D50 (1994) 5262; D. G. Robertson

: ”Theory of Hadrons and Light-Front QCD”, World Scientific, Singapore (1995) 111-

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116; A. B. Bylev, V. A. Franke, E. V. Prokhvatilov : hep-th/9711022; K. Yamawaki

: hep-th/9707141.

[4] J. B. Kogut, D. E. Soper : Phys. Rev. D1 (1970) 2901; F. Rohrlich : Acta Phys.

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: Phys. Rev. D8 (1973) 2736; J. H. Ten Eyck, F. Rohrlich : Phys. Rev. D9 (1974)

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[5] D. M. Capper, J. J. Dulwich, M. J. Litvak : Nucl. Phys. 241B (1984) 463.

[6] S. Mandelstam : Nucl. Phys. 213B (1983) 149; G. Leibbrandt : Phys. Rev. D29

(1984) 1699.

[7] A. Bassetto, M. Dalbosco, I. Lazzizzera, R. Soldati : Phys. Rev. D31 (1985) 2012.

[8] A. Bassetto, M. Dalbosco, R. Soldati : Phys. Rev. D36 (1987) 3138; C. Becchi :

”Physical and Nonstandard Gauges”, Springer-Verlag, Berlin (1990) 177-184.

[9] G. McCartor, D. G. Robertson : Z. Phys. C62 (1994) 349, C68 (1995) 345.

[10] G. McCartor : Z. Phys. C41 (1988) 271.

[11] P. A. M. Dirac : ”Lectures on Quantum Mechanics”, Belfer Graduate School of

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M. Mukunda : ”Classical Dynamics - A Modern Perspective”, Wyley Interscience,

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[12] R. Casalbuoni : Nuovo Cim. 33A (1976) 115.

[13] C. Itzykson, J. B. Zuber : ”Quantum Field Theory”, McGraw-Hill Int. Ed., Singapore

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[14] A. Bassetto, G. Nardelli, R. Soldati : ”Yang-Mills Theories in Algebraic Non-

Covariant Gauges”, World Scientific, Singapore (1991).

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29

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